(e) In deep and intermediate water depths, the significant wave height obtained by the spectral analysis

using the above equation is usually greater than that from the wave train analysis. The zero-crossing period

from the spectral method is only an approximation, while the period associated with the largest wave energy

known as the *peak period T*p, can only be obtained via the spectral analysis. In the spectral representation

of swell waves, there is a single value of the peak period and wave energy decays at frequencies to either side.

The spectra for storm waves is sometimes multi-peaked. One peak (not always the highest) corresponds to

the swell occurring at lower frequencies. One and sometimes more peaks are associated with storm waves

occurring at comparatively higher frequencies. In a double-peaked spectra for storm waves, the zero-crossing

period generally occurs at higher frequencies than the peak period. In a multi-peaked spectrum, the zero-

crossing period is not a measure of the frequency where peak energy occurs.

(5) Relationships among H1/3, Hs, and Hm0 in shallow water.

(a) By conception, significant height is the average height of the third-highest waves in a record of time

period. By tradition, wave height is defined as the distance from crest to trough. Significant wave height Hs

can be estimated from a wave-by-wave analysis in which case it is denoted H1/3, but more often is estimated

from the variance of the record or the integral of the variance in the spectrum in which case it is denoted Hm0.

Therefore, Hs in Equation II-1-152 should be replaced with Hm0 when the latter definition of Hs is implied.

While H1/3 is a direct measure of Hs, Hm0 is only an estimate of the significant wave height which under many

circumstances is accurate. In general in deep water H1/3 and Hm0 are very close in value and are both

considered good estimates of Hs. All modern wave forecast models predict Hm0 and the standard output of

most wave gauge records is Hm0. Few routine field gauging programs actually compute and report H1/3 and

report as Hs with no indication of how it was derived. Where H1/3 and Hs are equivalent, this is of little

concern.

(b) Thompson and Vincent (1985) investigated how H1/3 and Hm0 vary in very shallow water near

breaking. They found that the ratio H1/3/Hm0 varied systematically across the surf zone, approaching a

maximum near breaking. Thompson and Vincent displayed the results in terms of a nomogram (Figure II-1-

40). For steep waves, H1/3/Hm0 increased from 1 to about 1.1, then decreased to less than 1 after breaking.

For low steepness waves, the ratio increased from 1 before breaking to as much as 1.3-1.4 at breaking, then

decreased afterwards. Thompson and Vincent explained this systematic variation in the following way. As

low steepness waves shoal prior to breaking, the wave shape systematically changes from being near

sinusoidal to a wave shape that has a very flat trough with a very pronounced crest. Although the shape of

the wave is significantly different from the sine wave in shallow water, the variance of the surface elevation

is about the same, it is just arranged over the wave length differently from a sine wave. After breaking, the

wave is more like a bore, and as a result the H1/3 can be smaller (by about 10 percent) than Hm0.

(c) The critical importance of this research is in interpreting wave data near the surf zone. It is of

fundamental importance for the engineer to understand what estimate of significant height he is using and

what estimate is needed. As an example, if the data from a gauge is actually Hm0 and the waves are near

breaking, the proper estimate of Hs is given by H1/3. Given the steepness and relative depth, H1/3 may be

estimated from Hm0 by Figure II-1-40. Numerically modelled waves near the surf zone are frequently

equivalent to Hm0. In this case, Hs will be closer to H1/3 and the nomogram should be used to estimate Hs.

(6) Parametric spectrum models.

(a) In general, the spectrum of the sea surface does not follow any specific mathematical form.

However, under certain wind conditions the spectrum does have a specific shape. A series of empirical

expressions have been found which can be fit to the spectrum of the sea surface elevation. These are called

follows.

II-1-88

Water Wave Mechanics

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